To solve the problem \( x - \int \frac{2}{x^2} \, dx \), we need to perform the integration and then simplify the expression.

1. First, let's integrate \(\int \frac{2}{x^2} \, dx\).

\[ \int \frac{2}{x^2} \, dx = \int 2x^{-2} \, dx \]

2. Use the power rule for integration, which states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration:

\[ \int 2x^{-2} \, dx = 2 \int x^{-2} \, dx \]
\[ = 2 \left( \frac{x^{-1}}{-1} \right) + C \]
\[ = -2x^{-1} + C \]
\[ = -\frac{2}{x} + C \]

3. Now, substitute the integrated result back into the original expression:

\[ x - \left( -\frac{2}{x} + C \right) \]

4. Simplify the expression:

\[ x + \frac{2}{x} - C \]

So, the final result is:

\[ x + \frac{2}{x} - C \]

Would you like more details or have any questions about this solution?

Here are some more questions you might want to explore:

1. How do you integrate \(\int x^n \, dx\)?
2. What are the properties of definite integrals?
3. How do you solve a differential equation?
4. What is the Fundamental Theorem of Calculus?
5. How do you find the area under a curve?

**Tip:** Always double-check your integration results by differentiating them to see if you obtain the original integrand.

Learn how to integrate the expression x - 2/x^2 step-by-step with detailed explanations and examples. This problem involves integration techniques such as the power rule and the constant of integration, relevant to college-level mathematics.

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